Answer to the Friday Puzzle!

2) Can you add 3 lines to the equation and make it correct to six decimal places?

If you have not tried to solve it, have a go now. For everyone else, the answer is after the break.

1)

2) Did you solve it? Any other solutions?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

I arrived at (2) wondering how one gets a decimal point in there in the first place. Then I realized that some sort of constant would do, from which it was a short leap to the known rational approximations of π.

The notation in (2) denotes long division as written in the UK, the US, and several other countries. As I commented on Friday, this is by far not universal. I was not familiar with this notation until coming to the US. It does feel rather unusual for me still.

In any case, a straightforward real, even irrational and downright transcendental puzzle this time, rather than a creative lateral one😉

I got both, but it took me absolutely ages (and the help of a friend) to remember that you could represent division like that. I realised quickly that you could turn the left hand side into pi with only 1 line, so it seemed like an obvious candidate. Then I looked up approximations of pi, and found 355/113 fairly easily, it just then took a while to work out how to show that with only 2 lines!

If you ask me, it is simply a historical approach that stuck. I don’t see much logic in that notation either – perhaps that’s why people dread long division so much. In Germany a : b is used to denote a divided by b, and I never heard that complaint there. Regardless of the specific operator symbol, that notation is just more congruent with the semantics.

I knew I was looking for a way of writing pi = 355/133 at an immediate glance, but couldn’t do it in 3 lines. I don’t consider the proposed solution as a valid notation for division. Who has ever seen this, in the times when they used it, on one side of the = sign? No of course not, you put the answer on top. So it does not actually act as a representation of the answer, and thus is not a notation. What it is, in fact, is a construction line to facilitate the arrangement necessary for carrying out the operation manually on paper. We can compare it with the arrangement and aid lines used for carrying out a long multiplication: that is not a notation for multiplication either (apart from the x sign you may place there).

I got the second one as stated but had a different solution to the first one:
+1 = 11 – 3 – 35/5
= 8 – 7
= 1

i.e. added a horizontal line to the one on the left to make it a plus sign, added a minus between 11 and 3 and between 3 and 35 on the right side, and finally added a divide by sign between the 35 and the 5 on the right side